265 research outputs found
Characteristic boundary value problems: estimates from H1 to L2
Motivated by the study of certain non linear free-boundary value problems for
hyperbolic systems of partial differential equations arising in
Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the
solution to certain boundary value problems, in the conormal Sobolev space
H1_tan, can be transformed into an L2 a priori estimate of the same problem
Two-Dimensional Vortex Sheets for the Nonisentropic Euler Equations: Nonlinear Stability
We show the short-time existence and nonlinear stability of vortex sheets for
the nonisentropic compressible Euler equations in two spatial dimensions, based
on the weakly linear stability result of Morando--Trebeschi (2008) [20]. The
missing normal derivatives are compensated through the equations of the
linearized vorticity and entropy when deriving higher-order energy estimates.
The proof of the resolution for this nonlinear problem follows from certain
\emph{a priori} tame estimates on the effective linear problem {in the usual
Sobolev spaces} and a suitable Nash--Moser iteration scheme.Comment: to appear in: J. Differential Equations 2018. arXiv admin note:
substantial text overlap with arXiv:1707.0267
Well-posedness of the linearized problem for contact MHD discontinuities
We study the free boundary problem for contact discontinuities in ideal
compressible magnetohydrodynamics (MHD). They are characteristic
discontinuities with no flow across the discontinuity for which the pressure,
the magnetic field and the velocity are continuous whereas the density and the
entropy may have a jump. Under the Rayleigh-Taylor sign condition on the jump of the normal derivative of the pressure satisfied
at each point of the unperturbed contact discontinuity, we prove the
well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD
flows.Comment: 40 page
Existence of approximate current-vortex sheets near the onset of instability
The paper is concerned with the free boundary problem for 2D current-vortex
sheets in ideal incompressible magneto-hydrodynamics near the transition point
between the linearized stability and instability. In order to study the
dynamics of the discontinuity near the onset of the instability, Hunter and
Thoo have introduced an asymptotic quadratically nonlinear integro-differential
equation for the amplitude of small perturbations of the planar discontinuity.
The local-in-time existence of smooth solutions to the Cauchy problem for such
amplitude equation was already proven, under a suitable stability condition.
However, the solution found there has a loss of regularity (of order two) from
the initial data. In the present paper, we are able to obtain an existence
result of solutions with optimal regularity, in the sense that the regularity
of the initial data is preserved in the motion for positive times
Reduction by λ –symmetries and σ –symmetries: a Frobenius approach
Different kinds of reduction for ordinary differential equations, such as λ –symmetry and σ –symmetry reductions, are recovered as particular cases of Frobenius reduction theorem for distribution of vector fields. This general approach provides some hints to tackle the reconstruction problem and to solve it under suitable assumptions on the distribution involved in the reduction process
Maximal degree variational principles and Liouville dynamics
AbstractLet M be smooth n-dimensional manifold, fibered over a k-dimensional submanifold B as π:M→B, and ϑ∈Λk(M); one can consider the functional on sections φ of the bundle π defined by ∫Dφ∗(ϑ), with D a domain in B. We show that for k=n−2 the variational principle based on this functional identifies a unique (up to multiplication by a smooth function) nontrivial vector field in M, i.e., a system of ODEs. Conversely, any vector field X on M satisfying X⌟dϑ=0 for some ϑ∈Λn−2(M) admits such a variational characterization. We consider the general case, and also the particular case M=P×R where one of the variables (the time) has a distinguished role; in this case our results imply that any Liouville (volume-preserving) vector field on the phase space P admits a variational principle of the kind considered here
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